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Tuesday, October 22, 2013

Truth-making and Reference-making: Appendix

Appendix

Some may have noticed that on my account, as it stands, bivalence will fail for empty nouns or noun-phrases. A false-maker for a sentence is defined as a reference-maker for its subject term which is not a reference-maker for its predicate term. If the subject term is empty, it has no reference, so false-makers are not defined for sentences of this kind. Neither are truth-makers, for a truth-maker for a sentence is defined as a reference-maker for its subject term which is also a reference-maker for its predicate term, for these sentences their subject terms have no reference-makers. If we require that every true sentence has a truth-maker and that every false sentence has a false-maker, such sentences will come out as neither true nor false.

Many, I take it, will unhappy with this result. To reinstate bivalence, I'll introduce the concept of a negative reference-maker, in analogy with the concept of a false-maker for a sentence. A negative reference-maker is something that makes it the case that a term does not refer to anything. In the case of nouns and predicates, we could also call them empty-makers, for they make nouns and predicates empty.

I will now define this concept, calling what I have previously called reference-makers positive reference makers. I say that a noun has the proper class V of all objects (excluding proper classes) as its negative reference-maker iff it has no positive reference-makers. Similarly for monadic predicates. For relations, I say that an n-ary relational has the proper class of all n-tuples as its reference maker if it has no n-tuple as a positive reference-maker.

We can now give revised definitions of false-makers for subject-predicate and relational sentences:

An object x is a false-maker for a subject-predicate sentence p iff x is a positive reference-maker for p's subject term but is is not a positive reference-maker for p's predicate term. The proper class V of all objects is a false-maker for a subject-predicate sentence p iff V is a negative reference-maker for p's subject term or p's predicate term (or both).

An ordered n-tuple o is a false-maker for an n-ary relational sentence p iff the objects ordered in o are each positive reference-makers for one of p's subject terms, but o is not a positive reference-maker for p's predicate term. A proper class C is a false-maker for an n-ary relational sentence p iff C is a negative reference-maker for one of p's subject terms, or for p's predicate term, or both.

With these definitions ready to hand, we can see that sentences containing empty nouns or empty predicates come out as false rather than truth-valueless. "Nessie [a.k.a. the Loch Ness Monster] exists" comes out as false because the term 'Nessie', being empty, has the proper class V of all objects as its negative reference-maker, and so "Nessie exists" has V as its false-maker. Moreover, since "Nessie exists" has V as its false-maker, "It's not the case that Nessie exists" has V as its truth-maker. On the other hand, "Nessie is non-existent" comes out as false--if we construe 'non-existent' as a real predicate, then since 'Nessie' has V as its negative reference-maker, "Nessie is non-existent" has V as its false-maker. To me this seems intuitively to be the correct result: 'Nessie' is empty, so nothing is true of its referent--not even that it is non-existent--for it has no referent, so there's no "it". Nothing could possibly be non-existent.

However, "It is not the case that Nessie is non-existent" is true--since "Nessie is non-existent" has V has its false-maker, "It is not the case that Nessie is non-existent"has V as its truth-maker. Yet though "It is not the case that Nessie is non-existent" is true, it does not follow that "Nessie exists" is true--given that 'Nessie' does not refer, "Nessie exists" cannot be true for reasons just explained. What this shows is that "x is non-F" and "it is not the case that x is F" are not in general equivalent expressions, for the former entails the latter, but not conversely: In some situations where its not the case that x is F, "x is non-F" can nevertheless be false due to the fact that there is no such thing as x. We thus have the following three results:

1. 'Exists' is a meaningful predicate which is necessarily true of everything;
2. 'non-existent' is a meaningful predicate which is necessarily true of nothing; and
3. there can still be false positive existential sentences and true negative existential sentences.